Quantum Phase Estimation and its problems - hwakのトリビアルな雑記集

In this series of two articles, I will explain the theory of quantum phase estimation and the precautions to be taken when performing it. Currently, research on NISQ computers is going downhill all over the world, and the realization of an error-tolerant quantum computer is highly anticipated. By doing so, we can finally run algorithms that demonstrate quantum superiority, such as Grover, quantum Fourier transform, Shor, and quantum phase estimation algorithms. Among them, quantum phase estimation is a very promising algorithm in the field of quantum chemistry. This algorithm uses a quantum computer to calculate the eigenvalues and eigenvectors of a state and find its phase. This algorithm can be applied to the number of ancillary bits t as $O(t ^2)$ for the number of ancillary bits t. This algorithm requires a large number of ancillary bits to calculate the phase in addition to the computation bits. If the operator for which we want to calculate the eigenvalues and eigenvectors is $\exp(iHt)$ and we use n qubits, the ancillary bits will be t bits, and the initial state is,

$\mid 0 \rangle \mid \text{ini} \rangle$ -(1)

where $\mid \text{ini} \rangle$ is the initial state of n qubits. By applying H gate for all bits of ancilla bits, we obtain

$\sum_{j=0}^{2^t -1} \frac{1}{2^{t/2}} \mid j \rangle \mid \text{ini} \rangle$ -(2)

Then multiply $\exp(-iHt)$ by that value of ancilla bits, using the ancilla bits as control bits we obtain,

$\sum_{j=0}^{2^t -1} \frac{1}{2^{t/2}} \mid j \rangle U^j \mid \text{ini} \rangle$ -(3)

Any vector can be expressed as a sum of eigenvectors. Therefore, consider the eigenvectors of U as $\mid \phi_k \rangle$ and

$\sum_{k=0}^{2^n -1} c_k \sum_{j=0}^{2^t -1} \frac{1}{2^{t/2}} \mid j \rangle \lambda_k^j \mid \phi_k \rangle$ -(4)

Assuming $\lambda_k$ as $\exp(i \omega_k)$ , we can derive

$\sum_{k=0}^{2^n -1} c_k \mid \phi_k \rangle \sum_{j=0}^{2^t -1} \frac{1}{2^{t/2}} \exp(i \omega_k j) \mid j \rangle$ -(5)

The inverse quantum Fourier transform (IQFT) of the ancillary bits can easily calculate the eigenstates, and the final result is

$\sum_{k=0}^{2^n -1} c_k \mid \phi_k \rangle \mid \omega_k \rangle$ -(6)

Here, the phase $\omega_k$ is expressed as a binary fraction. Therefore, the accuracy in this algorithm is determined by the number of ancillary bits t. This sequence of events is represented by the circuit in Figure 1.

Figure 1: Circuit for quantum phase estimation.

Here is the main issue. The number of ancillary bits determines the accuracy of the algorithm. The equation for determining how many ancillary bits are needed to achieve the desired accuracy is given by the following equation, where the required accuracy is $1/2 ^a$ and the probability that the phase can be calculated with that accuracy is

$p(a, t) = 1 - \frac{1}{2^{(2^t - a - 2)}} = 1 - \epsilon(a, t)$ -(7)

Here, $\epsilon$ is the error probability with precision index a and the number of ancillary bits t. Figure 2 shows the normal logarithm of $\epsilon$ with respect to a and t. As shown in Figure 2, the probability that the phase and eigenvalues can be calculated with double precision is only possible with 55 qubits.

Figure 2: Relationship between the number of ancillary bits t and the error rate ϵ.

This means that quantum phase estimation is still difficult to achieve even by simulation. blueqat is limited to simulating 20 qubits, so even hydrogen is limited to single precision. In the second part, we will actually calculate and measure the accuracy and computation time.

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